In order to get the daily quests that can yield this pet, you need to complete the main quest chain in Deepholm. One of these quests is Pebble, which introduces you to this cutie (quest may be skippable). Once you’ve ‘made friends’ with Therazane, several daily quests become available in the throne area (and a portal between the throne and the temple becomes available), and three random dailies become available.
Pebble is an award for completing Rock Lover, which requires you to complete Lost In The Deeps ten times. You should get the quest for the first time immediately upon completing the main Deepholm quest chain. After that, it’s random, but some A3-ers have shared their top secret tips.
Park yourself in front of the quest giver, then start joining custom garrison groups (there is an understanding with those that each player is on his or her own; don’t join actual ‘working’ groups for this) until the ‘!’ quest marker shows up. After you accept the quest, leave your party. (Note: there is a nearby quest giver with 2-3 quests that you can easily do while you’re getting Pebble back… again.) (It was thought that having Kura was a guarantee, but that was either a major coincidence or changed with patch 6.2.)
To find Pebble, enter the Crumbling Wastes. You can use ground mounts inside, but you might get unmounted once or twice near the entrance. Go forward until you reach a turn. Do not go there yet. Wait for the huge elite Gyreworm to pass. Run behind it as closely as you can without touching it. There will be an opening with a dead end to your right, then, a little further ahead, a larger cavern with several halls opening off it. Run into this area. Take the first right out from this room, then the first left. You should find Pebble standing at the edge of a small ledge. Interact with him and get him to follow you. Normally, you can jump off this ledge and then move ahead until you reach the giant worm’s path again. Wait for it to pass, and turn right. Jump over the pile of rocks (Pebble will jump right over it) and run out. Do not fly until the Return Pebble to Safety’ completion notice pops up. Then you can return to the throne and turn in the quest. After doing this ten times, you will get Pebble as a reward. And, let’s face it, he might be safer with you.
Pebble is a little buggy. If you lose him, exit the cave to reset and go back in. He’ll be waiting again. Likewise, if the elite worm gets you, Pebble will be back where he was when you found him earlier.
Notes: you can do this quest once per day, like all other dailies. You may not find the quest marker with the suggested methods, so it may take longer than ten days to finish the achievement, but it shouldn’t take the months that others have had to endure.